Convolutions of random measures on compact groups

LetG be a compact group andM ₁(G) be the convolution semigroup of all Borel probability measures onG with the weak topology. We consider a stationary sequence {μ _n } _n=?∞ ^+∞ of random measures μ _n =μ _n (ω) inM ₁(G) and the convolutions $$v_{m,n} (\omega ) = \mu _m (\omega )* \cdots *\mu _{n - 1} (\omega ), m< n$$ and $$\alpha _n^{( + k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n,n + i} (\omega ),} \alpha _n^{( - k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n - i,n} (\omega )} $$ We describe the setsA _m ⁺ (ω) andA _n ⁺ (ω) of all limit points ofv _m,n(ω) asm→?∞ orn→+∞ and the setA _∞(ω) of its two-sided limit points for typical realizations of {μ _n (ω)} _n=?∞ ^+∞ . Using an appropriate random ergodic theorem we study the limit random measures ρ _n ^(±) (ω)=lim _k→∞ α _n ^(±k) (ω).     

Hausdorff measures for a class of homogeneous cantor sets

We consider the homogeneous Cantor sets which are generalization of symmetric perfect sets, and give a formula of the exact Hausdorff measures for a class of such sets.     

Convolutions of random measures on a compact topological semigroup

We investigate the asymptotic behavior of a sequence of convolutionsv ⁽ⁿ ₁() ** _n (), where { _n } _n=1 is some random process taking values in a semigroupM ¹(S) of probability Borel measures on a compact topological semigroupS.     

Convolutions of singular measures and applications to the Zakharov system

Uniform L²-estimates for the convolution of singular measures with respect to transversal submanifolds are proved in arbitrary space dimension. The results of Bennett-Bez are used to extend previous work of Bejenaru-Herr-Tataru. As an application, it is shown that the 3D Zakharov system is locally well-posed in the full subcritical regime.     

Interpretability of the cantor varieties

An SC-theory for a variety V is the collection of all strong Mal'tsev conditions satisfied in V. We prove that the SC-theory for the Cantor variety C ₂,defined in the signature {n, l, r}by three identities ln(x, y) =x, rn(x, y) =y, and n(lx, rx) =x, has a basis (i.e., an independent generating set) of any finite length. Translated fromAlgebra i Logika, Vol. 34, No. 4, pp. 464–471, July-August, 1995.     

Continuity of Stochastic Convolutions

Let B be a Brownian motion, and let be the space of all continuous periodic functions f with period 1. It is shown that the set of all f such that the stochastic convolution does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category.     

Stable intersections of affine cantor sets

We construct open sets of pairs of affine Cantor sets (K, K′) with the simplest possible combinatorics (i. e., defined by two affine increasing maps) which have stable intersection, while the product of lateral thicknesses τ_R(K) • τ_L (K′) is smaller than one. Thus, in a strong form, the converse to the following classic result due to Newhouse is not true: if the product of the thicknesses of two Cantor sets is bigger than one then there are translations of them which have stable intersection. We also describe the topological structure of K—K′ for typical pairs (K, K′) in this open set.     

Convolutions of Singular Distribution Functions

We construct an example that demonstrates that natural restrictions imposed on finite convolutions of singular distributions do not guarantee the purity of the limit distribution.     

两个齐次Cantor集的算术和

§ 1　IntroductionThe book[1 ] and the references therein show thatthe structure of arithmetic sums ofCantor sets is relevantto natural questions in smooth dynamics.Palis and Takens[1 ] askedabout the structure of the sums of two Cantor sets and conjectured that“typically” theyhave either zero Lebesgue measure or contained intervals. In 1 997,Solomyka[2 ] showedthatfor eachγ∈ 0 ,12 ,the set Kγ+Kλ(where Kλ,Kγis the middle-α Cantorset forα=1 -2λ or 1 -2γ) of two centered Cantor s…     

Exponential Integrability of Stochastic Convolutions

Sufficient conditions are found for stochastic convolution integralsdriven by a Wiener process in a Hilbert space to belong to theOrlicz space exp L²; standard exponential tail estimates followfrom these results. Proofs are based on the extrapolation theoryand are rather simple.     

非对称Cantor集的Hausdorff中心测度

§ 1　 IntroductionThe class of Cantor sets is a typical one of sets in fractal geometry.Mathematicianshave paid their attentions to such sets for a long time.Itis well known that the Hausdorffmeasure of the Cantor middle- third set is1(see[1]) .Recently,Feng[3] obtained the exactvalues of the packing measure for a class of linear Cantor sets.Using Feng s method,Zhuand Zhou[5] obtained the exactvalue of Hausdorff centred measure of the symmetry Cantorsets.In this papar,we consider the Ha…     

On hausdorff dimension of some cantor attractors

We study what happens with the dimension of Feigenbaum-like attractors of smooth unimodal maps as the order of the critical point grows.